14.7: W. A. Mozart- His Life and Legacy
Introduction
Wolfgang Amadeus Mozart (27 January 1756–5 December 1791), baptized as Johannes Chrysostomus Wolfgangus Theophilus Mozart, was a prolific and influential composer of the classical era. Born in Salzburg, Mozart showed prodigious ability from his earliest childhood. Already competent on keyboard and violin, he composed from the age of five and performed before European royalty. At 17, he was engaged as a musician at the Salzburg court, but grew restless and travelled in search of a better position. While visiting Vienna in 1781, he was dismissed from his Salzburg position. He chose to stay in the capital, where he achieved fame but little financial security. During his final years in Vienna, he composed many of his best-known symphonies, concertos, and operas, and portions of the Requiem, which was largely unfinished at the time of his death. The circumstances of his early death have been much mythologized. He was survived by his wife Constanze and two sons.
He composed more than 600 works, many acknowledged as pinnacles of symphonic, concertante, chamber, operatic, and choral music. He is among the most enduringly popular of classical composers, and his influence on subsequent Western art music is profound; Ludwig van Beethoven composed his own early works in the shadow of Mozart, and Joseph Haydn wrote that “posterity will not see such a talent again in 100 years.”
Early Life
Family and Childhood
Wolfgang Amadeus Mozart was born on 27 January 1756 to Leopold Mozart (1719–1787) and Anna Maria, née Pertl (1720–1778), at 9 Getreidegassein Salzburg. This was the capital of the Archbishopric of Salzburg, an ecclesiastic principality in what is now Austria, then part of the Holy Roman Empire. He was the youngest of seven children, five of whom died in infancy. His elder sister was Maria Anna (1751–1829), nicknamed “Nannerl.” Mozart was baptized the day after his birth at St. Rupert’s Cathedral. The baptismal record gives his name in Latinized form as Joannes Chrysostomus Wolfgangus Theophilus Mozart . He generally called himself “Wolfgang Amadè Mozart”as an adult, but his name had many variants.
Leopold Mozart, a native of Augsburg, was a minor composer and an experienced teacher. In 1743, he was appointed as fourth violinist in the musical establishment of Count Leopold Anton von Firmian, the ruling Prince-Archbishop of Salzburg. Four years later, he married Anna Maria in Salzburg. Leopold became the orchestra’s deputy Kapellmeister in 1763. During the year of his son’s birth, Leopold published a violin textbook, Versuch einer gründlichen Violinschule , which achieved success.
When Nannerl was seven, she began keyboard lessons with her father while her three-year-old brother looked on. Years later, after her brother’s death, she reminisced:
He often spent much time at the clavier, picking out thirds, which he was ever striking, and his pleasure showed that it sounded good…. In the fourth year of his age his father, for a game as it were, began to teach him a few minuets and pieces at the clavier…. He could play it faultlessly and with the greatest delicacy, and keeping exactly in time…. At the age of five, he was already composing little pieces, which he played to his father who wrote them down.
These early pieces, K. 1–5, were recorded in the Nannerl Notenbuch .
There is some scholarly debate of whether Mozart was four or five years old when he created his first musical compositions, though there is little doubt that Mozart composed his first three pieces of music within a few weeks of each other: KVs 1a, 1b and 1c.
Solomon notes that, while Leopold was a devoted teacher to his children, there is evidence that Mozart was keen to progress beyond what he was taught. His first ink-spattered composition and his precocious efforts with the violin were of his own initiative and came as a surprise to his father. Leopold eventually gave up composing when his son’s musical talents became evident. In his early years, Mozart’s father was his only teacher. Along with music, he taught his children languages and academic subjects.
Travel (1762–73)
During Mozart’s youth, his family made several European journeys in which he and Nannerl performed as child prodigies. These began with an exhibition, in 1762, at the court of the Prince-elector Maximilian III of Bavaria in Munich, and at the Imperial Court in Vienna and Prague. A long concert tour spanning three and a half years followed, taking the family to the courts of Munich, Mannheim, Paris, London, The Hague, again to Paris, and back home via Zurich, Donaueschingen, and Munich.
Mozart wrote his first symphony when he was eight years old. It is probable that his father transcribed most of it for him.
During this trip, Mozart met a number of musicians and acquainted himself with the works of other composers. A particularly important influence was Johann Christian Bach, whom Mozart visited in London in 1764 and 1765. The family again went to Vienna in late 1767 and remained there until December 1768.
These trips were often difficult and travel conditions were primitive. The family had to wait for invitations and reimbursement from the nobility and they endured long, near-fatal illnesses far from home: first Leopold (London, summer 1764), then both children (The Hague, autumn 1765).
After one year in Salzburg, Leopold and Mozart set off for Italy, leaving Mozart’s mother and sister at home. This travel lasted from December 1769 to March 1771. As with earlier journeys, Leopold wanted to display his son’s abilities as a performer and a rapidly maturing composer. Mozart met Josef Mysliveček and Giovanni Battista Martini in Bologna and was accepted as a member of the famous Accademia Filarmonica . In Rome, he heard Gregorio Allegri’s Miserere twice in performance in the Sistine Chapel and wrote it out from memory, thus producing the first unauthorized copy of this closely guarded property of the Vatican.
In Milan, Mozart wrote the opera Mitridate, re di Ponto (1770), which was performed with success. This led to further opera commissions. He returned with his father later twice to Milan (August–December 1771; October 1772 – March 1773) for the composition and premieres of Ascanio in Alba (1771) and Lucio Silla (1772). Leopold hoped these visits would result in a professional appointment for his son in Italy, but these hopes were never realized.
Toward the end of the final Italian journey, Mozart wrote the first of his works to be still widely performed today, the solo motet Exsultate, jubilate , K. 165.
Employment at the Salzburg Court (1773–77)
After finally returning with his father from Italy on 13 March 1773, Mozart was employed as a court musician by the ruler of Salzburg, Prince-Archbishop Hieronymus Colloredo. The composer had a great number of friends and admirers in Salzburg and had the opportunity to work in many genres, including symphonies, sonatas, string quartets, masses, serenades, and a few minor operas. Between April and December 1775, Mozart developed an enthusiasm for violin concertos, producing a series of five (the only ones he ever wrote), which steadily increased in their musical sophistication. The last three—K. 216, K. 218, K. 219—are now staples of the repertoire.
In 1776 Mozart turned his efforts to piano concertos, culminating in the E-flat concerto K. 271 of early 1777, considered by critics to be a breakthrough work.
Despite these artistic successes, Mozart grew increasingly discontented with Salzburg and redoubled his efforts to find a position elsewhere. One reason was his low salary, 150 florins a year; Mozart longed to compose operas, and Salzburg provided only rare occasions for these. The situation worsened in 1775 when the court theater was closed, especially since the other theater in Salzburg was largely reserved for visiting troupes.
Two long expeditions in search of work interrupted this long Salzburg stay: Mozart and his father visited Vienna from 14 July to 26 September 1773, and Munich from 6 December 1774 to March 1775. Neither visit was successful, though the Munich journey resulted in a popular success with the premiere of Mozart’s opera La finta giardiniera .
Journey to Paris (1777–78)
In August 1777, Mozart resigned his position at Salzburg and on 23 September ventured out once more in search of employment, with visits to Augsburg, Mannheim, Paris, and Munich. Mozart became acquainted with members of the famous orchestra in Mannheim, the best in Europe at the time. He also fell in love with Aloysia Weber, one of four daughters of a musical family. There were prospects of employment in Mannheim, but they came to nothing, and Mozart left for Paris on 14 March 1778 to continue his search. One of his letters from Paris hints at a possible post as an organist at Versailles, but Mozart was not interested in such an appointment. He fell into debt and took to pawning valuables. The nadir of the visit occurred when Mozart’s mother was taken ill and died on 3 July 1778. There had been delays in calling a doctor—probably, according to Halliwell, because of a lack of funds. Mozart stayed with Melchior Grimm, who, as personal secretary of the Duke d’Orléans, lived in his mansion.
While Mozart was in Paris his father was pursuing opportunities of employment for him in Salzburg. With the support of the local nobility, Mozart was offered a post as court organist and concertmaster. The annual salary was 450 florins, but he was reluctant to accept. By that time relations between Grimm and Mozart had cooled, and Mozart moved out. After leaving Paris in September 1778 for Strasbourg, he lingered in Mannheim and Munich, still hoping to obtain an appointment outside Salzburg. In Munich he again encountered Aloysia, now a very successful singer, but she was no longer interested in him. Mozart finally returned to Salzburg on 15 January 1779 and took up his new appointment, but his discontent with Salzburg remained undiminished.
Among the better-known works that Mozart wrote on the Paris journey are the A minor piano sonata, K. 310/300d and the “Paris” Symphony (No. 31), which were performed in Paris on 12 and 18 June 1778.
Vienna (1781–91)
Departure (1781)
In January 1781, Mozart’s opera Idomeneo premiered with “considerable success” in Munich. The following March, Mozart was summoned to Vienna, where his employer, Archbishop Colloredo, was attending the celebrations for the accession of Joseph II to the Austrian throne. Fresh from the adulation he had earned in Munich, Mozart was offended when Colloredo treated him as a mere servant and particularly when the archbishop forbade him to perform before the Emperor at Countess Thun’s for a fee equal to half of his yearly Salzburg salary. The resulting quarrel came to a head in May: Mozart attempted to resign and was refused. The following month, permission was granted but in a grossly insulting way: the composer was dismissed literally “with a kick in the arse,” administered by the archbishop’s steward, Count Arco. Mozart decided to settle in Vienna as a freelance performer and composer.
The quarrel with the archbishop went harder for Mozart because his father sided against him. Hoping fervently that he would obediently follow Colloredo back to Salzburg, Mozart’s father exchanged intense letters with his son, urging him to be reconciled with their employer. Mozart passionately defended his intention to pursue an independent career in Vienna. The debate ended when Mozart was dismissed by the archbishop, freeing himself both of his employer and his father’s demands to return. Solomon characterizes Mozart’s resignation as a “revolutionary step,” and it greatly altered the course of his life.
Early Years in Vienna
Mozart’s new career in Vienna began well. He performed often as a pianist, notably in a competition before the Emperor with Muzio Clementi on 24 December 1781, and he soon “had established himself as the finest keyboard player in Vienna”. He also prospered as a composer, and in 1782 completed the opera Die Entführung aus dem Serail (“The Abduction from the Seraglio”), which premiered on 16 July 1782 and achieved a huge success. The work was soon being performed “throughout German-speaking Europe,” and fully established Mozart’s reputation as a composer.
Near the height of his quarrels with Colloredo, Mozart moved in with the Weber family, who had moved to Vienna from Mannheim. The father, Fridolin, had died, and the Webers were now taking in lodgers to make ends meet. Aloysia, who had earlier rejected Mozart’s suit, was now married to the actor and artist Joseph Lange. Mozart’s interest shifted to the third Weber daughter, Constanze. The courtship did not go entirely smoothly; surviving correspondence indicates that Mozart and Constanze briefly separated in April 1782. Mozart faced a very difficult task in getting his father’s permission for the marriage. The couple were finally married on 4 August 1782 in St. Stephen’s Cathedral, the day before his father’s consent arrived in the mail.
The couple had six children, of whom only two survived infancy:
- Raimund Leopold (17 June–19 August 1783)
- Karl Thomas Mozart (21 September 1784–31 October 1858)
- Johann Thomas Leopold (18 October–15 November 1786)
- Theresia Constanzia Adelheid Friedericke Maria Anna (27 December 1787–29 June 1788)
- Anna Maria (died soon after birth, 16 November 1789)
- Franz Xaver Wolfgang Mozart (26 July 1791–29 July 1844)
In the course of 1782 and 1783, Mozart became intimately acquainted with the work of Johann Sebastian Bach and George Frideric Handel as a result of the influence of Gottfried van Swieten, who owned many manuscripts of the Baroque masters. Mozart’s study of these scores inspired compositions in Baroque style, and later influenced his personal musical language, for example in fugal passages in Die Zauberflöte (“The Magic Flute”) and the finale of Symphony No. 41.
In 1783, Mozart and his wife visited his family in Salzburg. His father and sister were cordially polite to Constanze, but the visit prompted the composition of one of Mozart’s great liturgical pieces, the Mass in C minor. Though not completed, it was premiered in Salzburg, with Constanze singing a solo part.
Mozart met Joseph Haydn in Vienna around 1784, and the two composers became friends. When Haydn visited Vienna, they sometimes played together in an impromptu string quartet. Mozart’s six quartets dedicated to Haydn (K. 387, K. 421, K. 428, K. 458, K. 464, and K. 465) date from the period 1782 to 1785, and are judged to be a response to Haydn’s Opus 33 set from 1781. Haydn in 1785 told Mozart’s father: “I tell you before God, and as an honest man, your son is the greatest composer known to me by person and repute, he has taste and what is more the greatest skill in composition.”
From 1782 to 1785 Mozart mounted concerts with himself as soloist, presenting three or four new piano concertos in each season. Since space in the theaters was scarce, he booked unconventional venues: a large room in the Trattnerhof (an apartment building), and the ballroom of the Mehlgrube (a restaurant). The concerts were very popular, and the concertos he premiered at them are still firm fixtures in the repertoire. Solomon writes that during this period Mozart created “a harmonious connection between an eager composer-performer and a delighted audience, which was given the opportunity of witnessing the transformation and perfection of a major musical genre”.
With substantial returns from his concerts and elsewhere, Mozart and his wife adopted a rather plush lifestyle. They moved to an expensive apartment, with a yearly rent of 460 florins. Mozart bought a fine fortepiano from Anton Walter for about 900 florins, and a billiard table for about 300. The Mozarts sent their son Karl Thomas to an expensive boarding school, and kept servants. Saving was therefore impossible, and the short period of financial success did nothing to soften the hardship the Mozarts were later to experience.
On 14 December 1784, Mozart became a Freemason, admitted to the lodge Zur Wohltätigkeit (“Beneficence”). Freemasonry played an important role in the remainder of Mozart’s life: he attended meetings, a number of his friends were Masons, and on various occasions he composed Masonic music, e. g. the Maurerische Trauermusik.
Return to Opera (1786–87)
Despite the great success of Die Entführung aus dem Serail , Mozart did little operatic writing for the next four years, producing only two unfinished works and the one-act Der Schauspieldirektor . He focused instead on his career as a piano soloist and writer of concertos. Around the end of 1785, Mozart moved away from keyboard writing and began his famous operatic collaboration with the librettist Lorenzo Da Ponte. 1786 saw the successful premiere of The Marriage of Figaro in Vienna. Its reception in Prague later in the year was even warmer, and this led to a second collaboration with Da Ponte: the opera Don Giovanni , which premiered in October 1787 to acclaim in Prague, but less success in Vienna in 1788. The two are among Mozart’s most important works and are mainstays of the operatic repertoire today, though at their premieres their musical complexity caused difficulty for both listeners and performers. These developments were not witnessed by Mozart’s father, who had died on 28 May 1787.
In December 1787, Mozart finally obtained a steady post under aristocratic patronage. Emperor Joseph II appointed him as his “chamber composer,” a post that had fallen vacant the previous month on the death of Gluck. It was a part-time appointment, paying just 800 florins per year, and required Mozart only to compose dances for the annual balls in the Redoutensaal. This modest income became important to Mozart when hard times arrived. Court records show that Joseph’s aim was to keep the esteemed composer from leaving Vienna in pursuit of better prospects.
In 1787 the young Ludwig van Beethoven spent several weeks in Vienna, hoping to study with Mozart. No reliable records survive to indicate whether the two composers ever met.
Later Years and Death
Financial Decline (1787–90)
Toward the end of the decade, Mozart’s circumstances worsened. Around 1786 he had ceased to appear frequently in public concerts, and his income shrank. This was a difficult time for musicians in Vienna because of the Austro-Turkish War: both the general level of prosperity and the ability of the aristocracy to support music had declined.
By mid-1788, Mozart and his family had moved from central Vienna to the suburb of Alsergrund. Although it has been thought that Mozart reduced his rental expenses, research shows that by moving to the suburb, Mozart had not reduced his expenses (as claimed in his letter to Puchberg), but merely increased the housing space at his disposal. Mozart began to borrow money, most often from his friend and fellow Mason Michael Puchberg; “a pitiful sequence of letters pleading for loans” survives. Maynard Solomon and others have suggested that Mozart was suffering from depression, and it seems that his output slowed. Major works of the period include the last three symphonies (Nos. 39, 40, and 41, all from 1788), and the last of the three Da Ponte operas, Così fan tutte , premiered in 1790.
Around this time, Mozart made long journeys hoping to improve his fortunes: to Leipzig, Dresden, and Berlin in the spring of 1789, and to Frankfurt, Mannheim, and other German cities in 1790. The trips produced only isolated success and did not relieve the family’s financial distress.
Last Works (1791)
Mozart’s last year was, until his final illness struck, a time of great productivity—and by some accounts, one of personal recovery. He composed a great deal, including some of his most admired works: the opera The Magic Flute ; the final piano concerto (K. 595 in B-flat); the Clarinet Concerto K. 622; the last in his great series of string quintets (K. 614 in E-flat); the motet Ave verum corpus K. 618; and the unfinished Requiem K. 626.
Mozart’s financial situation, a source of extreme anxiety in 1790, finally began to improve. Although the evidence is inconclusive, it appears that wealthy patrons in Hungary and Amsterdam pledged annuities to Mozart in return for the occasional composition. He is thought to have benefited from the sale of dance music written in his role as Imperial chamber composer. Mozart no longer borrowed large sums from Puchberg, and made a start on paying off his debts.
He experienced great satisfaction in the public success of some of his works, notably The Magic Flute (which was performed several times in the short period between its premiere and Mozart’s death) and the Little Masonic Cantata K. 623, premiered on 15 November 1791.
Final Illness and Death
Mozart fell ill while in Prague for the 6 September 1791 premiere of his opera La clemenza di Tito , written in that same year on commission for the Emperor’s coronation festivities. He continued his professional functions for some time, and conducted the premiere of The Magic Flute on 30 September. His health deteriorated on 20 November, at which point he became bedridden, suffering from swelling, pain, and vomiting.
Mozart was nursed in his final illness by his wife and her youngest sister, and was attended by the family doctor, Thomas Franz Closset. He was mentally occupied with the task of finishing his Requiem, but the evidence that he actually dictated passages to his student Franz Xaver Süssmayr is minimal.
Mozart died in his home on 5 December 1791 (aged 35) at 1:00 am. The New Grove describes his funeral:
Mozart was interred in a common grave, in accordance with contemporary Viennese custom, at the St. Marx Cemetery outside the city on 7 December. If, as later reports say, no mourners attended, that too is consistent with Viennese burial customs at the time; later Jahn (1856) wrote that Salieri, Süssmayr, van Swieten and two other musicians were present. The tale of a storm and snow is false; the day was calm and mild.
The expression “common grave” refers to neither a communal grave nor a pauper’s grave, but to an individual grave for a member of the common people (i.e., not the aristocracy). Common graves were subject to excavation after ten years; the graves of aristocrats were not.
The cause of Mozart’s death cannot be known with certainty. The official record has it as “hitziges Frieselfieber” (“severe miliary fever,” referring to a rash that looks like millet seeds), more a description of the symptoms than a diagnosis. Researchers have posited at least 118 causes of death, including acute rheumatic fever, streptococcal infection, trichinosis, influenza, mercury poisoning, and a rare kidney ailment.
Mozart’s modest funeral did not reflect his standing with the public as a composer: memorial services and concerts in Vienna and Prague were well-attended. Indeed, in the period immediately after his death, his reputation rose substantially: Solomon describes an “unprecedented wave of enthusiasm” for his work; biographies were written (first by Schlichtegroll, Niemetschek, and Nissen); and publishers vied to produce complete editions of his works.
Appearance and Character
Mozart’s physical appearance was described by tenor Michael Kelly, in his Reminiscences : “a remarkably small man, very thin and pale, with a profusion of fine, fair hair of which he was rather vain”. His early biographer Niemetschek wrote, “there was nothing special about [his] physique. […] He was small and his countenance, except for his large intense eyes, gave no signs of his genius.” His facial complexion was pitted, a reminder of his childhood case of smallpox. He loved elegant clothing. Kelly remembered him at a rehearsal: “he was on the stage with his crimson pelisse and gold-laced cocked hat, giving the time of the music to the orchestra.” Of his voice his wife later wrote that it “was a tenor, rather soft in speaking and delicate in singing, but when anything excited him, or it became necessary to exert it, it was both powerful and energetic”.
Mozart usually worked long and hard, finishing compositions at a tremendous pace as deadlines approached. He often made sketches and drafts; unlike Beethoven’s these are mostly not preserved, as his wife sought to destroy them after his death.
He was raised a Catholic and remained a loyal member of the Church throughout his life.
Mozart lived at the center of the Viennese musical world, and knew a great number and variety of people: fellow musicians, theatrical performers, fellow Salzburgers, and aristocrats, including some acquaintance with the Emperor Joseph II. Solomon considers his three closest friends to have been Gottfried von Jacquin, Count August Hatzfeld, and Sigmund Barisani; others included his older colleague Joseph Haydn, singers Franz Xaver Gerl and Benedikt Schack, and the horn player Joseph Leutgeb. Leutgeb and Mozart carried on a curious kind of friendly mockery, often with Leutgeb as the butt of Mozart’s practical jokes.
He enjoyed billiards and dancing, and kept pets: a canary, a starling, a dog, and a horse for recreational riding. He had a startling fondness for scatological humor, which is preserved in his surviving letters, notably those written to his cousin Maria Anna Thekla Mozart around 1777–1778, and in his correspondence with his sister and parents. Mozart also wrote scatological music, a series of canons that he sang with his friends.
Contributors and Attributions
- Revision and adaptation. Provided by : Lumen Learning and Natalia Kuznetsova. License : CC BY-SA: Attribution-ShareAlike
- Wolfgang Amadeus Mozart. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Wolfgang_Amadeus_Mozart . License : CC BY-SA: Attribution-ShareAlike
- Mozart's Old Home. Authored by : Mbah Gondrong. Located at : https://en.Wikipedia.org/wiki/File:Mozart%27s_old_home.jpg . License : CC BY-SA: Attribution-ShareAlike
- Mozartkugel. Authored by : Maria. Located at : https://www.flickr.com/photos/87735595@N04/16230784612/ . License : CC BY-ND: Attribution-NoDerivatives
- The Boy Mozart. Provided by : Mozarteum, Salzburg. Located at : https://en.Wikipedia.org/wiki/File:Wolfgang-amadeus-mozart_2.jpg . License : Public Domain: No Known Copyright
- Mozart's Birthplace. Authored by : Jonathan White. Located at : https://en.Wikipedia.org/wiki/File:Mozart_(5).JPG . License : Public Domain: No Known Copyright
- Portrait de Wolfgang Amadeus Mozart . Provided by : Google Cultural Institute. Located at : https://en.Wikipedia.org/wiki/File:Louis_Carrogis_dit_Carmontelle_-_Portrait_de_Wolfgang_Amadeus_Mozart_(Salzbourg,_1756-Vienne,_1791)_jouant_%C3%A0_Paris_avec_son_p%C3%A8re_Jean..._-_Google_Art_Project.jpg . License : Public Domain: No Known Copyright
- Portrait of Mozart (1756-1791). Provided by : Wikimedia. Located at : https://en.Wikipedia.org/wiki/File:Martini_bologna_mozart_1777.jpg . License : Public Domain: No Known Copyright
- Family Portrait. Provided by : Schiller Institute. Located at : https://en.Wikipedia.org/wiki/File:Wolfgang01.jpg . License : Public Domain: No Known Copyright
- Portrait of Constanze Mozart (1762-1842). Provided by : Mozart Forum. Located at : https://en.Wikipedia.org/wiki/File:Costanze_Mozart_by_Lange_1782.jpg . License : Public Domain: No Known Copyright
- Silverpoint Portrait of Wolfgang Amadeus Mozart. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/File:Mozart_drawing_Doris_Stock_1789.jpg . License : Public Domain: No Known Copyright
- Wolfgang Amadeus Mozart. Authored by : Barbara Krafft. Located at : https://en.Wikipedia.org/wiki/File:Wolfgang-amadeus-mozart_1.jpg . License : Public Domain: No Known Copyright
- Unfinished Portrait of Mozart. Authored by : Joseph Lange. Provided by : Mozart Forum. Located at : https://en.Wikipedia.org/wiki/File:Mozart_(unfinished)_by_Lange_1782.jpg . License : Public Domain: No Known Copyright


Speaking involves controlling parts of the mouth and nose to shape the air that comes from the lungs. This unit introduces the names and locations of the articulators that are used to produce the sounds of English: the tongue, lips and teeth, the alveolar ridge, the palate, the velum, and the nasal cavity.
A YouTube element has been excluded from this version of the text. You can view it online here: ecampusontario.pressbooks.pu...guistics/?p=40
Check Yourself
1. Which articulators are responsible for the first sound in the word
minor
?
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Lips.
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Lips and teeth.
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Tongue and teeth.
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Tongue and alveolar ridge.
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Tongue and palate.
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Tongue and velum.
2. Which articulators are responsible for the final sound in the word
wit
?
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Lips.
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Lips and teeth.
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Tongue and teeth.
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Tongue and alveolar ridge.
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Tongue and palate.
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Tongue and velum.
3. Which articulators are responsible for the first sound in the word
photography
?
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Lips.
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Lips and teeth.
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Tongue and teeth.
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Tongue and alveolar ridge.
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Tongue and palate.
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Tongue and velum.
Answers


16.5A: Mutualism vs. Symbiosis
- Compare Mutualism and Symbiosis
Symbiosis is any relationship between two or more biological species. Such relationships are usually long term and have a strong impact on the fitness of one or both organisms. Symbiotic relationships are categorized by the benefits and physical relationships experienced by each species.
Common types of symbiosis are categorized by the degree to which each species benefits from the interaction:
- Mutualism: In mutualistic interactions, both species benefit from the interaction. A classic example of mutualism is the relationship between insects that pollinate plants and the plants that provide those insects with nectar or pollen. Another classic example is the behavior of mutualistic bacteria in ecology and human health. Gut bacteria in particular are very important for digestion in humans and other species. In humans, gut bacteria assist in breaking down additional carbohydrates, out-competing harmful bacteria, and producing hormones to direct fat storage. Humans lacking healthy mutualistic gut flora can suffer a variety of diseases, such as irritable bowel syndrome. Some ruminant animals, like cows or deer, rely on special mutualistic bacteria to help them break down the tough cellulose in the plants they eat. In return, the bacteria get a steady supply of food.
- Commensalism: In commensalism, one organism benefits while the other organism neither benefits nor suffers from the interaction. For example, a spider may build a web on a plant and benefit substantially, while the plant remains unaffected. Similarly, a clown fish might live inside a sea anemone and receive protection from predators, while the anemone neither benefits nor suffers.
- Parasitism: Parasites are organisms that harm their symbiotic partners. Parasitism is incredibly common in nature: depending on the definition, more than half of all species may go through at least one parasitic stage in their life cycle. There are many well-documented examples of parasitic bacteria and microorganisms throughout this text.
Symbiosis can also be characterized by an organism’s physical relationship with its partner.
- Endosymbiosis: a relationship in which one of the symbiotic species lives inside the tissue the other. For example, Coral polyps have special algae called zooxanthelle that live inside their cells. Zooxanthelle provide sugars to the coral through photosynthesis. Similarly, nitrogen-fixing fungi often live inside the cells of plants, providing nitrogen in exchange for the sugars of photosynthesis.
- Ectosymbiosis: a relationship in which one species lives on the outside surface of the other. Barnacles that live on whales and bromeliads that live on tropical trees are examples of endosymbionts.
These categories can be paired with the above terms to better describe the species’ interactions. For example, you might say that a gut bacteria is an “endosymbiotic mutualist,” or that a flea is an “ectosymbiotic parasite. ”
Key Points
- Mutualism, a relationship in which both species benefit, is common in nature. In microbiology, there are many examples of mutualistic bacteria in the gut that aid digestion in both humans and animals.
- Commensalism is a relationship between species in which one benefits and the other is unaffected. Humans are host to a variety of commensal bacteria in their bodies that do not harm them but rely on them for survival (e.g. bacteria that consume dead skin).
- Parasitic relationships, in which one species benefits and the other suffers, are very common in nature. Most of the microorganisms studied in medical microbiology are parasitic and feed on human tissue. For example, cholera, leshmaniasis, and Giardia are all parasitic microbes.
- Symbiotic relationships can also be classified by the physical relationship between the two species. Endosymbionts live inside the tissues of the host, while ectosymbionts live outside of their partner species.
Key Terms
- commensalism : A class of relationship between two organisms in which one organism benefits without affecting the other
- symbiosis : A close and often long-term interaction between two or more different biological species
- mutualism : A relationship between individuals of different species in which both individuals benefit


3.2A: Components of Plasma Membranes
The plasma membrane protects the cell from its external environment, mediates cellular transport, and transmits cellular signals.
- Describe the function and components of the plasma membrane
Key Points
- The principal components of the plasma membrane are lipids ( phospholipids and cholesterol), proteins, and carbohydrates.
- The plasma membrane protects intracellular components from the extracellular environment.
- The plasma membrane mediates cellular processes by regulating the materials that enter and exit the cell.
- The plasma membrane carries markers that allow cells to recognize one another and can transmit signals to other cells via receptors.
Key Terms
- plasma membrane : The semipermeable barrier that surrounds the cytoplasm of a cell.
- receptor : A protein on a cell wall that binds with specific molecules so that they can be absorbed into the cell.
Structure of Plasma Membranes
The plasma membrane (also known as the cell membrane or cytoplasmic membrane) is a biological membrane that separates the interior of a cell from its outside environment.
The primary function of the plasma membrane is to protect the cell from its surroundings. Composed of a phospholipid bilayer with embedded proteins, the plasma membrane is selectively permeable to ions and organic molecules and regulates the movement of substances in and out of cells. Plasma membranes must be very flexible in order to allow certain cells, such as red blood cells and white blood cells, to change shape as they pass through narrow capillaries.
The plasma membrane also plays a role in anchoring the cytoskeleton to provide shape to the cell, and in attaching to the extracellular matrix and other cells to help group cells together to form tissues. The membrane also maintains the cell potential.
In short, if the cell is represented by a castle, the plasma membrane is the wall that provides structure for the buildings inside the wall, regulates which people leave and enter the castle, and conveys messages to and from neighboring castles. Just as a hole in the wall can be a disaster for the castle, a rupture in the plasma membrane causes the cell to lyse and die.
The plasma membrane : The plasma membrane is composed of phospholipids and proteins that provide a barrier between the external environment and the cell, regulate the transportation of molecules across the membrane, and communicate with other cells via protein receptors.
The Plasma Membrane and Cellular Transport
The movement of a substance across the selectively permeable plasma membrane can be either “passive”—i.e., occurring without the input of cellular energy —or “active”—i.e., its transport requires the cell to expend energy.
The cell employs a number of transport mechanisms that involve biological membranes:
- Passive osmosis and diffusion: transports gases (such as O 2 and CO 2) and other small molecules and ions
- Transmembrane protein channels and transporters: transports small organic molecules such as sugars or amino acids
- Endocytosis: transports large molecules (or even whole cells) by engulfing them
- Exocytosis: removes or secretes substances such as hormones or enzymes
The Plasma Membrane and Cellular Signaling
Among the most sophisticated functions of the plasma membrane is its ability to transmit signals via complex proteins. These proteins can be receptors, which work as receivers of extracellular inputs and as activators of intracellular processes, or markers, which allow cells to recognize each other.
Membrane receptors provide extracellular attachment sites for effectors like hormones and growth factors, which then trigger intracellular responses. Some viruses, such as Human Immunodeficiency Virus (HIV), can hijack these receptors to gain entry into the cells, causing infections.
Membrane markers allow cells to recognize one another, which is vital for cellular signaling processes that influence tissue and organ formation during early development. This marking function also plays a later role in the “self”-versus-“non-self” distinction of the immune response. Marker proteins on human red blood cells, for example, determine blood type (A, B, AB, or O).


7.3: Musical Scales
Introduction
The first thing musicians must do before they can play together is "tune.” For musicians in the standard Western music tradition, this means agreeing on exactly what pitch (what frequency) is an "A,” what is a "B flat," and so on. Other cultures not only have different note names and different scales, they may even have different notes - different pitches - based on a different tuning system. In fact, the modern Western tuning system, which is called equal temperament, replaced (relatively recently) other tuning systems that were once popular in Europe. All tuning systems are based on the physics of sound. But they all are also affected by the history of their music traditions, as well as by the tuning peculiarities of the instruments used in those traditions.
Many musical instruments, including pianos, guitars, and violins, make sound when strings are plucked, hit, or set in vibration in some other way. This vibration causes the air to move, creating an audible wave, which eventually reaches our ears. We call this wave “sound.” The pitch—how high or low the sound is—is determined by various factors such as the length of the string, its thickness, and its tension. A guitarist may play an open string, say G, and then place a finger on a fret to create sound with a higher pitch, say G#, A, or B. What is happening is that the player is shortening the vibrating part of the string to produce a higher pitch. This is based on the fundamental fact that shorter (or skinner or tighter) strings produce higher pitches. Similar facts are true with woodwind, percussion, and other types of instruments in general: the smaller the vibrating portion gets, the higher the sound becomes. That is why, for instance, a tuba is bigger than a French horn, and a contrabass flute is longer than a soprano flute.
In this section, we focus our attention on string vibrations. In particular, we look at the relation between the pitch and the string length, assuming that the material, thickness, and tightness of the string remain constant.
Suppose we pluck a string of a certain length and create a wave, a vibration of the air. The string will move up and down in oscillation. Each wave, whether it is a simple water wave, sound, earthquake, radio wave, microwave, or light wave, has a frequency, which measures how many times the oscillation occurs in a given time period, such as one second. For sound, the frequency is measured by “cycles per second” (abbreviated “cps”), sometimes called “hertz” (denoted “Hz”). Frequency is what determines the pitch of the sound generated by the vibration.
When a string is plucked and set in motion, the entire string vibrates, as shown in the picture above. The frequency created by the whole string is referred to as the “fundamental frequency” of the string. It generates the lowest sound the string can possibly make. This is equivalent to playing an open string on the guitar. If the guitar is tuned in the standard way, the sixth string (the thickest one, located on the top) is the E string, meaning that its fundamental frequency corresponds to the pitch we call an “E.” This is the lowest sound a guitar can play.
Now, if we take that same string, press the midpoint of it and pluck it, only half of the string would vibrate, shortening the wave length to half and doubling the frequency. This, of course, results in a higher pitch. But how much higher? When the frequency is doubled, we call the interval between those two pitches an “octave” (meaning 8). For instance, if the open string has a frequency of 440 cps, then the half string would have a frequency of 880 cps, and that sound is called an octave higher than the first. One can, of course, keep making the vibrating portion shorter and shorter to create higher and higher sounds.
Pythagoras: Numbers, Music, and Harmony
Although music has existed in different cultures for millennia, it was Pythagoras, an ancient Greek mathematician and philosopher, who explicitly linked sound (pitch) with frequency (number), laying a foundation of music theory based on mathematics. It was known to Pythagoras that two notes (sounds with different frequencies) sound nice together (harmonious, pleasant) when the ratio of the two frequencies is a simple fraction. For instance, two pitches that are an octave apart have the ratio of 2 to 1 as described above. 2/1 (or its reciprocal ½) is a simple fraction, and these two notes sound nice if played simultaneously. In fact, if two people sing a song one octave apart, it could sound almost like a unison. These simple fractions give rise to well-known interval names in music theory:
2 = 2.00: octave
5/3 = 1.666…: major sixth
3/2 = 1.50: perfect fifth
4/3 = 1.333…: perfect fourth
5/4 = 1.25: major third
Pythagoras once said, "There is geometry in the humming of the strings, there is music in the spacing of the spheres."
These simple ratios became a foundation for various musical scales and musical chords. For this reason, sometimes Pythagoras is called the father of music theory. In fact, as a religious and philosophical leader, he taught that ratios create harmony which forms beauty in nature; therefore, ratios are divine. His followers, called the Pythagoreans, thus did not believe the existence of numbers that are not ratios of integers. It is then quite ironic that Pythagoras himself later proved that square root of 2 (the length of the hypotenuse of a right isosceles triangle whose two sides are 1 unit long) is irrational, contradicting his own teaching that every number is rational. (By the way, this number, 2, came about by a very theorem named after him—the Pythagorean Theorem!)
Regardless of this devastating discovery, numerical ratios played a very significant role in the composition of musical scales. A scale is an ordered list of music tones that covers an octave; for instance, today the standard C-major scale consists of 8 notes starting with a C and ending with the next high C. I am certain that readers who are piano students have practiced these scales thousands of times (and your family may be quite tired of hearing them).
Throughout history, there have been many musical scales in various cultures, traditions, genres, and generations. Many authors have documented a wide variety of scales and methods of tuning instruments. Interested readers can look up topics such as Pythagorean intonation, mean-tone system, and temperaments. For simplicity, we now focus on the current music scale structure, consisting of 12 notes and their frequencies. This scale may not have been so widespread (or so standard) until J. S. Bach around the beginning of the 18th century.
Here are some facts about today’s standard musical scale:
- A scale begins and ends with two pitches with the same name, one octave apart (meaning that the frequency of the last note is exactly twice that of the first note).
- This whole interval, an octave, is divided into 12 half-steps, in such a way that the frequency ratio of any consecutive half-steps is the same (described as “equal distance apart”).
Let’s take a look at these requirements mathematically now.
Suppose \(P_0\) represents the frequency of the first note in a scale. Every time you move half-step to the next note, there must be a fixed ratio increase, i.e., growth by a fixed percentage, say, \(r\) (percentage written as a decimal). So the second note has the frequency
\[P_{1}=P_{0}(1+r)\]
Then, to go to the next note by moving half-step up, we need to multiply this new number by the same multiplier, giving us
\[P_{2}=P_{1}(1+r)=P_{0}(1+r)(1+r)=P_{0}(1+r)^{2}\]
Repeating, we get
\[P_{3}=P_{0}(1+r)^{3}\]
\[ P_{4}=P_{0}(1+r)^{4}\]
and so on. This look quite familiar, correct? This is what we saw in exponential growth. This is what we saw in compounding interest. Yes, it turns out that the mathematics that rules musical scales is the same as the mathematics that describes compound interest in finance and any exponential (fixed-percentage) growth in natural or social science!
So what is this magic number \(r\), the common ratio that separates every half-step on the musical scale? Remember that after 12 half-steps, we have traveled an octave, doubling the original frequency. In other words, this is the number r so that the following equation is satisfied:
\[P_{12}=P_{0}(1+r)^{12}=2 P_{0} \nonumber \]
If this equation were to appear in the context of finance, then this r would be the annual interest rate such that the original principal is doubled after 12 years. After dividing both sides by P0 and taking the 12th root of both sides, we get
\[1+r=\sqrt[12]{2} \approx 1.059463 \nonumber \]
We finally conclude that the fixed ratio between any two half-steps is approximately 0.059463, which is close to 5.946%. So, if you get this interest rate in a bank account compounding annually, you will double your balance in exactly 12 years. If you start your musical scale with a note with a certain frequency, after 12 half-steps, you will double your frequency, resulting in the note exactly one octave higher. (Think of the first two notes in the song “Somewhere Over the Rainbow.”) Below is a graph showing what happens to your bank balance if you begin with the original principal of $440 with the annual rate 5.946% for 24 years; your money doubles twice, i.e., quadruples to $1,760. This is like hitting a note two octaves higher.
Why did I start this graph with $440? Well, in standard tuning, the middle A is set to 440 cps. Tuning forks are designed to vibrate at this frequency (if you know what a tuning fork is). Electronic tuners allow you to adjust this a bit, depending on the music you play (or how you want your music to sound). Increasing this frequency, 440 cps, by \(r=0.059463\) each time (for 12 times), you will get the following notes in the standard A scale. For some notes, the ratio of the frequency and 440 is also shown.
|
Note |
Frequency (Hz) |
Ratio |
|
|---|---|---|---|
|
A |
440.00 |
1 |
|
|
A# |
466.16 |
||
|
B |
493.88 |
1.122462 |
|
|
C |
523.25 |
1.189207 |
|
|
C# |
554.37 |
1.259921 |
approx. 5:4 |
|
D |
587.33 |
1.33484 |
approx. 4:3 |
|
D# |
622.25 |
||
|
E |
659.26 |
1.498307 |
approx. 3:2 |
|
F |
698.46 |
||
|
F# |
739.99 |
||
|
G |
783.99 |
||
|
G# |
830.61 |
||
|
A |
880.00 |
2 |
exactly 2:1 |
Take a look, for instance, at C#, with a frequency ratio of 1.2599, which is approximately 5:4. This note in the key of A major is called “major third”; D is called “perfect fourth,” and E is “perfect fifth.” Because of these simple (approximate) ratios, the notes A, C#, and E sound beautiful together. The set of these three notes is called the “A major” chord. All harmony is based on these simple ratios.
But each ratio is irrational and not exactly a simple ratio, right? This is the unfortunate result of equal temperament (devised in the 18th century); the common ratio r is the (not-so-secret) number that would allow a piece of music to sound good in every key, but the trade-off of temperament is that all ratios are now approximate, except for octaves, and thus, strictly speaking, “out of tune.” However, given the difficulty in tuning big instruments like the piano, this was the preferred compromise. The alternative is to have some ratios perfect while others are way off simple ratios, resulting in some music perhaps sounding great in one key but sounding terrible in another. Equal temperament is thus widely accepted and regarded as standard now. In other words, one irrational number, 122, seems to control the harmonious world of music.
Just one more thing… I mentioned that the A major chord consists of A, C#, and E. If you change that middle note and lower it by half-step, you get the notes A, C (natural), and E, which together form the A minor chord. Play these chords and hear the difference. The mood shifts dramatically, right? Most people can hear the difference between a song written in a major key and a minor key. In fact, it would make people laugh if you take a popular song written in a minor key and play it in the corresponding major key. That huge difference, we now know, is generated by one little note dropped by one half-step, i.e., a frequency difference of about 5.946%.
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0


5.7: Theory of Plate Tectonics
When the concept of seafloor spreading came along, scientists recognized that it was the mechanism to explain how continents could move around Earth’s surface. Like the scientists before us, we will now merge the ideas of continental drift and seafloor spreading into the theory of plate tectonics.
Watch this video about continental drift and the mechanism of seafloor spreading create plate tectonics.
EARTH’S TECTONIC PLATES
Seafloor and continents move around on Earth’s surface, but what is actually moving? What portion of the Earth makes up the “plates” in plate tectonics? This question was also answered because of technology developed during war times – in this case, the Cold War. The plates are made up of the lithosphere.
During the 1950s and early 1960s, scientists set up seismograph networks to see if enemy nations were testing atomic bombs. These seismographs also recorded all of the earthquakes around the planet. The seismic records could be used to locate an earthquake’s epicenter , the point on Earth’s surface directly above the place where the earthquake occurs.
Figure 1. Earthquakes outline the plates.
Earthquake epicenters outline the plates. Mid-ocean ridges, trenches, and large faults mark the edges of the plates, and this is where earthquakes occur (figure 1).
The lithosphere is divided into a dozen major and several minor plates (figure 2). The plates’ edges can be drawn by connecting the dots that mark earthquakes’ epicenters. A single plate can be made of all oceanic lithosphere or all continental lithosphere, but nearly all plates are made of a combination of both.
Figure 2. The lithospheric plates and their names. The arrows show whether the plates are moving apart, moving together, or sliding past each other.
Movement of the plates over Earth’s surface is termed plate tectonics . Plates move at a rate of a few centimeters a year, about the same rate fingernails grow.
HOW PLATES MOVE
If seafloor spreading drives the plates, what drives seafloor spreading? Picture two convection cells side-by-side in the mantle, similar to the illustration in figure 3.
Figure 3. Mantle convection drives plate tectonics. Hot material rises at mid-ocean ridges and sinks at deep sea trenches, which keeps the plates moving along the Earth’s surface.
- Hot mantle from the two adjacent cells rises at the ridge axis, creating new ocean crust.
- The top limb of the convection cell moves horizontally away from the ridge crest, as does the new seafloor.
- The outer limbs of the convection cells plunge down into the deeper mantle, dragging oceanic crust as well. This takes place at the deep sea trenches.
- The material sinks to the core and moves horizontally.
- The material heats up and reaches the zone where it rises again.
Check out this animation of mantle convection and watch this video:
PLATE BOUNDARIES
Plate boundaries are the edges where two plates meet. Most geologic activities, including volcanoes, earthquakes, and mountain building, take place at plate boundaries. How can two plates move relative to each other?
- Divergent plate boundaries : the two plates move away from each other.
- Convergent plate boundaries : the two plates move towards each other.
- Transform plate boundaries : the two plates slip past each other.
The type of plate boundary and the type of crust found on each side of the boundary determines what sort of geologic activity will be found there.
Divergent Plate Boundaries
Plates move apart at mid-ocean ridges where new seafloor forms. Between the two plates is a rift valley. Lava flows at the surface cool rapidly to become basalt, but deeper in the crust, magma cools more slowly to form gabbro.  Earthquakes are common at mid-ocean ridges since the movement of magma and oceanic crust results in crustal shaking. The vast majority of mid-ocean ridges are located deep below the sea (figure 4).
Figure 4. (a) Iceland is the one location where the ridge is located on land: the Mid-Atlantic Ridge separates the North American and Eurasian plates; (b) The rift valley in the Mid-Atlantic Ridge on Iceland.
Check out these animations:
- Divergent plate boundary at mid-ocean ridge
- Divergent plate boundary
Can divergent plate boundaries occur within a continent? What is the result? Incontinental rifting (figure 5), magma rises beneath the continent, causing it to become thinner, break, and ultimately split apart. New ocean crust erupts in the void, creating an ocean between continents.
Figure 5. The Arabian, Indian, and African plates are rifting apart, forming the Great Rift Valley in Africa. The Dead Sea fills the rift with seawater.
Convergent Plate Boundaries
When two plates converge, the result depends on the type of lithosphere the plates are made of. No matter what, smashing two enormous slabs of lithosphere together results in magma generation and earthquakes.
Figure 6. Subduction of an oceanic plate beneath a continental plate causes earthquakes and forms a line of volcanoes known as a continental arc.
Ocean-Continent
When oceanic crust converges with continental crust, the denser oceanic plate plunges beneath the continental plate. This process, called subduction , occurs at the oceanic trenches (figure 6). The entire region is known as a subduction zone . Subduction zones have a lot of intense earthquakes and volcanic eruptions. The subducting plate causes melting in the mantle. The magma rises and erupts, creating volcanoes. These coastal volcanic mountains are found in a line above the subducting plate (figure 7). The volcanoes are known as a continental arc .
Figure 7. (a) At the trench lining the western margin of South America, the Nazca plate is subducting beneath the South American plate, resulting in the Andes Mountains (brown and red uplands); (b) Convergence has pushed up limestone in the Andes Mountains where volcanoes are common.
The movement of crust and magma causes earthquakes. Look at this map of earthquake epicenters at subduction zones. This animation shows the relationship between subduction of the lithosphere and creation of a volcanic arc.
The volcanoes of northeastern California—Lassen Peak, Mount Shasta, and Medicine Lake volcano—along with the rest of the Cascade Mountains of the Pacific Northwest are the result of subduction of the Juan de Fuca plate beneath the North American plate (figure 8). The Juan de Fuca plate is created by seafloor spreading just offshore at the Juan de Fuca ridge.
Figure 8. The Cascade Mountains of the Pacific Northwest are a continental arc.
If the magma at a continental arc is felsic, it may be too viscous (thick) to rise through the crust. The magma will cool slowly to form granite or granodiorite. These large bodies of intrusive igneous rocks are called batholiths , which may someday be uplifted to form a mountain range (figure 9).
Figure 9. The Sierra Nevada batholith cooled beneath a volcanic arc roughly 200 million years ago. The rock is well exposed here at Mount Whitney. Similar batholiths are likely forming beneath the Andes and Cascades today.
Ocean-Ocean
When two oceanic plates converge, the older, denser plate will subduct into the mantle. An ocean trench marks the location where the plate is pushed down into the mantle. The line of volcanoes that grows on the upper oceanic plate is an island arc . Do you think earthquakes are common in these regions (figure 10)?
Figure 10. (a) Subduction of an ocean plate beneath an ocean plate results in a volcanic island arc, an ocean trench and many earthquakes. (b) Japan is an arc-shaped island arc composed of volcanoes off the Asian mainland, as seen in this satellite image.
Check out this animation of an ocean continent plate boundary.
Continent-Continent
Continental plates are too buoyant to subduct. What happens to continental material when it collides? Since it has nowhere to go but up, this creates some of the world’s largest mountains ranges (figure 11). Magma cannot penetrate this thick crust so there are no volcanoes, although the magma stays in the crust. Metamorphic rocks are common because of the stress the continental crust experiences. With enormous slabs of crust smashing together, continent-continent collisions bring on numerous and large earthquakes.
Figure 11. (a) In continent-continent convergence, the plates push upward to create a high mountain range. (b) The world’s highest mountains, the Himalayas, are the result of the collision of the Indian Plate with the Eurasian Plate, seen in this photo from the International Space Station.
Check out this short animation of the Indian Plate colliding with the Eurasian Plate.
Watch this animation of the Himalaya rising.
The Appalachian Mountains are the remnants of a large mountain range that was created when North America rammed into Eurasia about 250 million years ago.
Transform Plate Boundaries
Transform plate boundaries are seen as transform faults , where two plates move past each other in opposite directions. Transform faults on continents bring massive earthquakes (figure 12).
Figure 12. At the San Andreas Fault in California, the Pacific Plate is sliding northeast relative to the North American plate, which is moving southwest. At the northern end of the picture, the transform boundary turns into a subduction zone.
California is very geologically active. What are the three major plate boundaries in or near California (figure 13)?
- A transform plate boundary between the Pacific and North American plates creates the San Andreas Fault, the world’s most notorious transform fault.
- Just offshore, a divergent plate boundary, Juan de Fuca ridge, creates the Juan de Fuca plate.
- A convergent plate boundary between the Juan de Fuca oceanic plate and the North American continental plate creates the Cascades volcanoes.
Figure 13. This map shows the three major plate boundaries in or near California.
A brief review of the three types of plate boundaries and the structures that are found there is the subject of this wordless video.
EARTH’S CHANGING SURFACE
Geologists know that Wegener was right because the movements of continents explain so much about the geology we see. Most of the geologic activity that we see on the planet today is because of the interactions of the moving plates.
Figure 14. Mountain ranges of North America.
In the map of North America (figure 14), where are the mountain ranges located? Using what you have learned about plate tectonics, try to answer the following questions:
- What is the geologic origin of the Cascades Range? The Cascades are a chain of volcanoes in the Pacific Northwest. They are not labelled on the diagram but they lie between the Sierra Nevada and the Coastal Range.
- What is the geologic origin of the Sierra Nevada? (Hint: These mountains are made of granitic intrusions.)
- What is the geologic origin of the Appalachian Mountains along the Eastern US?
Remember that Wegener used the similarity of the mountains on the west and east sides of the Atlantic as evidence for his continental drift hypothesis. The Appalachian mountains formed at a convergent plate boundary as Pangaea came together (figure 15).
Figure 15. About 200 million years ago, the Appalachian Mountains of eastern North America were probably once as high as the Himalaya, but they have been weathered and eroded significantly since the breakup of Pangaea.
Before Pangaea came together, the continents were separated by an ocean where the Atlantic is now. The proto-Atlantic ocean shrank as the Pacific ocean grew. Currently, the Pacific is shrinking as the Atlantic is growing. This supercontinent cycle is responsible for most of the geologic features that we see and many more that are long gone (figure 16).
Figure 16. Scientists think that the creation and breakup of a supercontinent takes place about every 500 million years. The supercontinent before Pangaea was Rodinia. A new continent will form as the Pacific ocean disappears.
This animation shows the movement of continents over the past 600 million years beginning with the breakup of Rodinia.
LESSON SUMMARY
- Plates of lithosphere move because of convection currents in the mantle. One type of motion is produced by seafloor spreading.
- Plate boundaries can be located by outlining earthquake epicenters.
- Plates interact at three types of plate boundaries: divergent, convergent and transform.
- Most of the Earth’s geologic activity takes place at plate boundaries.
- At a divergent boundary, volcanic activity produces a mid ocean ridge and small earthquakes.
- At a convergent boundary with at least one oceanic plate, an ocean trench, a chain of volcanoes develops and many earthquakes occur.
- At a convergent boundary where both plates are continental, mountain ranges grow and earthquakes are common.
- At a transform boundary, there is a transform fault and massive earthquakes occur but there are no volcanoes.
- Processes acting over long periods of time create Earth’s geographic features.
REFLECTION QUESTIONS
- What skill does this content help you develop?
- What are the key topics covered in this content?
- How can the content in this section help you demonstrate mastery of a specific skill?
- What questions do you have about this content?
Contributors and Attributions
-
Original content from Kimberly Schulte (Columbia Basin College) and supplemented by Lumen Learning . The content on this page is copyrighted under a Creative Commons Attribution 4.0 International license.


17.1: Waves
Waves form on the ocean and on lakes because energy from the wind is transferred to the water. The stronger the wind, the longer it blows, and the larger the area of water over which it blows (the fetch ), the larger the waves are likely to be.
The important parameters of a wave are the wavelength (the horizontal distance between two crests or two troughs), the amplitude (the vertical distance between a trough and a crest ), and the wave velocity (the speed at which wave crests move across the water) (Figure \(\PageIndex{1}\)).
The typical sizes and speeds of waves in situations where they have had long enough to develop fully are summarized in Table 17.1. In a situation where the fetch is short (say 19 km on a lake) and the wind is only moderate (19 km/h), the waves will develop fully within 2 hours, but they will remain quite small (average amplitude about 27 cm, wavelength 8.5 m). On a large body of water (the ocean or a very large lake) with a fetch of 139 km and winds of 37 km/h, the waves will develop fully in 10 hours; the average amplitude will be around 1.5 m and average wavelength around 34 m. In the open ocean, with strong winds (92 km/h) that blow for at least 69 hours, the waves will average nearly 15 m high and their wavelengths will be over 200 m. Small waves (amplitudes under a metre) tend to have relatively shallow slopes (amplitude is 3% to 4% of wavelength), while larger waves (amplitudes over 10 m) have much steeper slopes (amplitude is 6% to 7% of wavelength). In other words, not only are large waves bigger than small ones, they are also generally more than twice as steep, and therefore many times more impressive—and potentially dangerous. It is important to recognize, however, that amplitudes decrease with distance from the area where the waves were generated. Waves on our coast that are generated by a storm near Japan will have similar wavelengths but lower amplitudes than those generated by a comparable storm just offshore.
| Wind Speed (kilometers per hour) | Fetch (kilometers) | Duration (hours) | Amplitude (meters) | Wavelength (meters) | Wave Period (seconds) | Wave Velocity (meters per second) | Wave Velocity (kilometers per hour) |
|---|---|---|---|---|---|---|---|
| 19 | 19 | 2 | 0.27 | 8.5 | 3.0 | 2.8 | 10.2 |
| 37 | 139 | 10 | 1.5 | 33.8 | 5.7 | 5.9 | 19.5 |
| 56 | 518 | 23 | 4.1 | 76.5 | 8.6 | 8.9 | 32.0 |
| 74 | 1,313 | 42 | 8.5 | 136 | 11.4 | 11.9 | 42.9 |
| 92 | 2,627 | 69 | 14.8 | 212 | 14.3 | 14.8 | 53.4 |
This table shows the typical amplitudes and wavelengths of waves generated under different wind conditions. The steepness of a wave can be determined from these numbers and is related to the ratio: amplitude/wavelength.
- Calculate these ratios for the waves shown. The first one is done for you.
- How would these ratios change with increasing distance from the wind that produced the waves?
| Amplitude (meters) | Wavelength (meters) | Ratio (amplitude divided by wavelength) |
|---|---|---|
| 0.27 | 8.5 | 0.03 |
| 1.5 | 33.8 | |
| 4.1 | 76.5 | |
| 8.5 | 136 | |
| 14.8 | 212 |
See Appendix 3 for Exercise 17.1 answers .
Relatively small waves move at up to about 10 km/h and arrive on a shore about once every 3 seconds. Very large waves move about five times faster (over 50 km/h), but because their wavelengths are so much longer, they arrive less frequently—about once every 14 seconds.
As a wave moves across the surface of the water, the water itself mostly just moves up and down and only moves a small amount in the direction of wave motion. As this happens, a point on the water surface describes a circle with a diameter that is equal to the wave amplitude (Figure \(\PageIndex{2}\)). This motion is also transmitted to the water underneath, and the water is disturbed by a wave to a depth of approximately one-half of the wavelength. Wave motion is illustrated quite clearly on the Wikipedia “Wind wave” site . If you look carefully at that animation, and focus on the small white dots in the water, you should be able to see how the amount that they move decreases with depth.
The one-half wavelength depth of disturbance of the water beneath a wave is known as the wave base . Since ocean waves rarely have wavelengths greater than 200 m, and the open ocean is several thousand meters deep, the wave base does not normally interact with the bottom of the ocean. However, as waves approach the much shallower water near the shore, they start to “feel” the bottom, and they are affected by that interaction (Figure \(\PageIndex{3}\)). The wave “orbits” are both flattened and slowed by dragging, and the implications are that the wave amplitude (height) increases and the wavelength decreases (the waves become much steeper). The ultimate result of this is that the waves lean forward, and eventually break (Figure \(\PageIndex{4}\)).
Waves normally approach the shore at an angle, and this means that one part of the wave feels the bottom sooner than the rest of it, so the part that feels the bottom first slows down first. This process is illustrated in Figure \(\PageIndex{5}\), which is based on an aerial photograph showing actual waves approaching Long Beach on Vancouver Island. When the photo was taken, the waves (with crests shown as white lines in the diagram) were approaching at an angle of about 20° to the beach. The waves first reached shore at the southern end (“a” on the image). As they moved into shallow water they were slowed, and since the parts of the waves still in deep water (“b” on the image) were not slowed they were able catch up, and thus the waves became more parallel to the beach.
In open water, these waves had wavelengths close to 100 m. In the shallow water closer to shore, the wavelengths decreased to around 50 m, and in some cases, even less.
Even though they bend and become nearly parallel to shore, most waves still reach the shore at a small angle, and as each one arrives, it pushes water along the shore, creating what is known as a longshore current within the surf zone (the areas where waves are breaking) (Figure \(\PageIndex{6}\)).
A series of waves (dashed lines) is approaching the coast on the map shown here. The location of the depth contour that is equivalent to 1/2 of the wavelength is shown as a red dashed line.
- Draw in the next several waves, showing how their patterns will change as they approach shallow water and the shore.
- Show, with arrows, the direction of the resulting longshore current.
See Appendix 3 for Exercise 17.2 answers .
Another important effect of waves reaching the shore at an angle is that when they wash up onto the beach, they do so at an angle, but when that same wave water flows back down the beach, it moves straight down the slope of the beach (Figure \(\PageIndex{8}\)). The upward-moving water, known as the swash , pushes sediment particles along the beach, while the downward-moving water, the backwash , brings them straight back. With every wave that washes up and then down the beach, particles of sediment are moved along the beach in a zigzag pattern.
The combined effects of sediment transport within the surf zone by the longshore current and sediment movement along the beach by swash and backwash is known as longshore drift . Longshore drift moves a tremendous amount of sediment along coasts (both oceans and large lakes) around the world, and it is responsible for creating a variety of depositional features that we’ll discuss in section 17.3.
A rip current is another type of current that develops in the nearshore area, and has the effect of returning water that has been pushed up to the shore by incoming waves. As shown in Figure \(\PageIndex{9}\), rip currents flow straight out from the shore and are fed by the longshore currents. They die out quickly just outside the surf zone, but can be dangerous to swimmers who get caught in them. If part of a beach does not have a strong unidirectional longshore current, the rip currents may be fed by longshore currents going in both directions.
Rip currents are visible in Figure \(\PageIndex{10}\), a beach at Tunquen in Chile near Valparaiso. As is evident from the photo, the rips correspond with embayments in the beach profile. Three of them are indicated with arrows, but it appears that there may be several others farther along the beach.
Tides are related to very long-wavelength but low-amplitude waves on the ocean surface (and to a much lesser extent on very large lakes) that are caused by variations in the gravitational effects of the Sun and Moon. Tide amplitudes in shoreline areas vary quite dramatically from place to place. On the west coast of Canada, the tidal range is relatively high, in some areas as much as 6 m, while on most of the east coast the range is lower, typically around 2 m. A major exception is the Bay of Fundy between Nova Scotia and New Brunswick, where the daily range can be as great as 16 m. Anomalous tides like that are related to the shape and size of bays and inlets, which can significantly enhance the amplitude of the tidal surge. The Bay of Fundy has a natural oscillation cycle of 12.5 hours, and that matches the frequency of the rise and fall of the tides in the adjacent Atlantic Ocean. Ungava Bay, on Quebec’s north coast, has a similarly high tidal range.
As the tides rise and fall they push and pull a large volume of water in and out of bays and inlets and around islands. They do not have as significant an impact on coastal erosion and deposition as wind waves do, but they have an important influence on the formation of features within the intertidal zone, as we’ll see in the following sections.
Media Attributions
- Figures 17.1.1, 17.1.2, 17.1.3, 17.1.6, 17.1.7, 17.1.8, 17.1.9: © Steven Earle. CC BY.
- Figure \(\PageIndex{10}\): “ Tunquen Chile ” by Cecilia and Randy Lascody. P ublic domain.
- Table by Steven Earle from data at: en.Wikipedia.org/wiki/Wind_wave] ↵


1.3: Week 3 - The Purpose of Argument (How to Be Wrong)
Why do we argue with one another? What is the intention? How do we know when we have “won” or “lost” an argument? What happens then? In this unit, we explore the functions and purposes of argument to reveal the deeper reasons we engage in this complex, frequently stressful activity and what we can gain by having an enhanced perspective on it.
Argument is not in itself an end or a purpose of communication. It is rather a means of discourse, a way of developing what we have to say. We can identify four primary aims or purposes that argument helps us accomplish:
- Inquiry
- Conviction
- Persuasion
- Negotiation
- View the Powerpoint: The Four Aims of Argument
Arguing to Inquire: Forming our opinions or questioning those we already have.
The ancient Greeks used the word dialectic to identify an argument as inquiry; a more common term might be dialogue or conversation. Arguing to inquire helps us accomplish the following:
- to form opinions
- to question opinions
- to reason our way through conflicts or contradictions
It requires an attitude of patient questioning under non-threatening circumstances, usually done alone or among trusted friends and associates. The primary purpose is a search for the truth. The primary audience is often the writer and fellow inquirers concerned with the same issues.
- Examples: Classroom discussions; journal writing; exploratory essays; letters; late-night bull sessions in a dorm.
Arguing to Convince: Gaining assent from others through case-making.
While some inquiry may be never-ending, the goal of most inquiry is to reach a conclusion, a conviction. We seek an earned opinion, achieved through careful thought, research, and discussion. And then we usually want others to share this conviction, to secure the assent of an audience by means of reason rather than by force.
- Arguing to inquire centers on asking questions: we want to expose and examine what we think.
- Arguing to convince requires us to make a case, to get others to agree with what we think. While inquiry is a cooperative use of argument, convincing is competitive. We put our case against the case of others in an effort to win the assent of readers.
- Examples: a lawyer’s brief; newspaper editorials; case studies; most academic writing
Arguing to Persuade: Moving others to action through rational, emotional, personal, and stylistic appeals.
While arguing to convince seeks to earn the assent of readers or listeners, arguing to persuade attempts to influence their behavior, to move them to act upon the conviction. Persuasion aims to close the gap between assent and action. To convince focuses on the logic of an argument; to persuade will often rely on the personal appeal of the writer (what Aristotle called ethos) and involve an appeal to an audience’s emotions (pathos). In addition to these personal and emotional appeals, persuasion exploits the resources of language more fully than convincing does.
- In general, the more academic the audience or the more purely intellectual the issue, the more likely that the writing task involves an argument to convince rather than to persuade. In most philosophy or science assignments, for example, the writer would usually focus on conviction rather than persuasion, confining the argument primarily to thesis, reasons, and evidence. But when you are working with public issues, with matters of policy or questions of right and wrong, persuasion’s fuller range of appeal is usually appropriate.
- Persuasion begins with difference and, when it works, ends with identity. We expect that before reading our argument, readers will differ from us in beliefs, attitudes, and/or desires. A successful persuasive argument brings readers and writer together, creating a sense of connection between parties.
- Examples: Political speeches, sermons, advertising
Arguing to Negotiate: Exploring differences of opinion in the hope of reaching agreement and/or cooperation.
If efforts to convince and/or persuade the audience have failed, the participants must often turn to negotiation, resolving the conflict in order to maintain a satisfactory working relationship.
- Each side must listen closely to understand the other side’s case and the emotional commitments and values that support that case. The aim of negotiation is to build consensus, usually by making and asking for concessions. Dialogue plays a key role, bringing us full circle back to argument as inquiry. Negotiation often depends on collaborative problem-solving.
- Examples: Diplomatic negotiations, labor relations, documents in organizational decision-making; essays seeking resolution of conflict between competing parties; also frequent in private life when dealing with disagreements among friends and family members.
The Importance of Being Wrong
In this module, you are reading articles and watching videos that explore the science and logic of why we argue and why being wrong is not something we should try to avoid at all costs or view as “losing.” Collaboration and clarification of ideas are the highest pursuits of argumentative communication and when we are proven wrong, we are given the opportunity to learn, to grow and to enhance our understanding of the complex and vibrant world we inhabit.
- View the video: For Argument’s Sake , https://www.youtube.com/watch?v=JTN9Nx8VYtk&feature=youtu.be
- View the video: On Being Wrong , https://www.youtube.com/watch?v=QleRgTBMX88&feature=youtu.be
Both of these talks (above) address the dangers of being too close to our own ideas. They offer examples and insights that show what can go wrong when we would rather “feel” right than “be” right. They also address the opportunities that open up to us when we allow ourselves to be detached enough from our ideas to create the space for growth, doubt, investigation and eventually increased understanding and awareness. When we believe we are right about everything all of the time, we miss the opportunities to learn from one another and from each new perspective we encounter. The following articles address the scientific basis for what embracing being “wrong” can actually offer our individual and collective ways of knowing.


12.5A: Brief Overview of Cranial Nerves
- Describe the function of the cranial nerves
Cranial nerves are the nerves that emerge directly from the brain (including the brainstem). In contrast, spinal nerves emerge from segments of the spinal cord. Cranial nerves relay information between the brain and parts of the body, primarily to and from regions of the head and neck.
Cranial Nerve Anatomy and Terminology
Spinal nerves emerge sequentially from the spinal cord with the spinal nerve closest to the head (C1) emerging in the space above the first cervical vertebra. The cranial nerves emerge from the central nervous system above this level.
Each cranial nerve is paired and is present on both sides. The numbering of the cranial nerves is based on the order in which they emerge from the brain, front to back (brainstem).
The terminal nerves, olfactory nerves (I) and optic nerves (II) emerge from the cerebrum or forebrain, and the remaining ten pairs arise from the brainstem, which is the lower part of the brain. The cranial nerves are considered components of the peripheral nervous system.
However, on a structural level, the olfactory, optic, and terminal nerves are more accurately considered part of the central nervous system.
The twelve cranial nerves are shown in the figure below followed by brief descriptions.
The cranial nerves : The locations of the cranial nerves within the brain.
- The olfactory nerve (I): This is instrumental for the sense of smell, it is one of the few nerves that are capable of regeneration.
- The optic nerve (II): This nerve carries visual information from the retina of the eye to the brain.
- The oculomotor nerve (III): This controls most of the eye’s movements, the constriction of the pupil, and maintains an open eyelid.
- The trochlear nerve (IV): A motor nerve that innervates the superior oblique muscle of the eye, which controls rotational movement.
- The trigeminal nerve (V): This is responsible for sensation and motor function in the face and mouth.
- The abducens nerve (VI): A motor nerve that innervates the lateral rectus muscle of the eye, which controls lateral movement.
- The facial nerve (VII): This controls the muscles of facial expression, and functions in the conveyance of taste sensations from the anterior two-thirds of the tongue and oral cavity.
- The vestibulocochlear nerve (VIII): This is responsible for transmitting sound and equilibrium (balance) information from the inner ear to the brain.
- The glossopharyngeal nerve (IX): This nerve receives sensory information from the tonsils, the pharynx, the middle ear, and the rest of the tongue.
- The vagus nerve (X): This is responsible for many tasks, including heart rate, gastrointestinal peristalsis, sweating, and muscle movements in the mouth, including speech and keeping the larynx open for breathing.
- The spinal accessory (XI): This nerve controls specific muscles of the shoulder and neck.
- The hypoglossal nerve (XII): This nerve controls the tongue movements of speech, food manipulation, and swallowing.
There are many mnemonic devices to remember the cranial nerves. One that may be helpful is: Old Opie Occasionally Tries Trigonometry And Feels Very Gloomy, Vague And Hypoactive.
Key Points
- The cranial nerves serve functions such as smell, sight, eye movement, and feeling in the face. The cranial nerves also control balance, hearing, and swallowing.

- The vagus nerve (X) has many branches and is responsible for tasks including heart rate, gastrointestinal peristalsis, sweating, and muscle movements in the mouth, including speech and keeping the larynx open for breathing.
Key Terms
- innervate : To supply an organ or other body part with nerves.
- peripheral nervous system : The nerves and ganglia outside of the brain and spinal cord.
- peristalsis : A radially symmetrical contraction and relaxation of muscles that propagates in a wave down a tube (such as the gastrointestinal tract) in an anterograde direction.


14.4E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{21}\) Determine if a Sequence is Geometric
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
- \(3,12,48,192,768,3072, \dots\)
- \(2,10,50,250,1250,6250, \dots\)
- \(48,24,12,6,3, \frac{3}{2}, \dots\)
- \(54,18,6,2, \frac{2}{3}, \frac{2}{9}, \dots\)
- \(-3,6,-12,24,-48,96, \dots\)
- \(2,-6,18,-54,162,-486, \dots\)
- Answer
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1. The sequence is geometric with common ratio \(r=4\).
3. The sequence is geometric with common ratio \(r=\frac{1}{2}\).
5. The sequence is geometric with common ratio \(r=−2\).
Exercise \(\PageIndex{22}\) Determine if a Sequence is Geometric
In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.
- \(48,24,12,6,3, \frac{3}{2}, \ldots\)
- \(12,6,0,-6,-12,-18, \dots\)
- \(-7,-2,3,8,13,18, \dots\)
- \(5,9,13,17,21,25, \ldots\)
- \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \dots\)
- \(4,8,12,24,48,96, \dots\)
- Answer
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1. The sequence is geometric with common ratio \(r=\frac{1}{2}\).
3. The sequence is arithmetic with common difference \(d=5\).
5. The sequence is geometric with common ratio \(r=\frac{1}{2}\).
Exercise \(\PageIndex{23}\) Determine if a Sequence is Geometric
In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.
- \(a_{1}=4\) and \(r=3\)
- \(a_{1}=9\) and \(r=2\)
- \(a_{1}=-4\) and \(r=-2\)
- \(a_{1}=-5\) and \(r=-3\)
- \(a_{1}=27\) and \(r=\frac{1}{3}\)
- \(a_{1}=64\) and \(r=\frac{1}{4}\)
- Answer
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1. \(4,12,36,108,324\)
3. \(-4,8,-16,32,-64\)
5. \(27,9,3,1, \frac{1}{3}\)
Exercise \(\PageIndex{24}\) Find the General Term (\(n\)th Term) of a Geometric Sequence
- Find \(a_{11}\) given \(a_{1}=8\) and \(r=3\).
- Find \(a_{13}\) given \(a_{1}=7\) and \(r=2\).
- Find \(a_{10}\) given \(a_{1}=-6\) and \(r=-2\).
- Find \(a_{15}\) given \(a_{1}=-4\) and \(r=-3\).
- Find \(a_{10}\) given \(a_{1}=100,000\) and \(r=0.1\).
- Find \(a_{8}\) given \(a_{1}=1,000,000\) and \(r=0.01\).
- Answer
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1. \(472,392\)
3. \(3,072\)
5. \(0.0001\)
Exercise \(\PageIndex{25}\) Find the General Term (\(n\)th Term) of a Geometric Sequence
In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.
- Find \(a_{9}\) of the sequence, \(9,18,36,72,144,288, \dots\)
- Find \(a_{12}\) of the sequence, \(5,15,45,135,405,1215, \dots\)
- Find \(a_{15}\) of the sequence, \(-486,162,-54,18,-6,2, \dots\)
- Find \(a_{16}\) of the sequence, \(224,-112,56,-28,14,-7, \ldots\)
- Find \(a_{10}\) of the sequence, \(1,0.1,0.01,0.001,0.0001,0.00001, \ldots\)
- Find \(a_{9}\) of the sequence, \(1000,100,10,1,0.1,0.01, \dots\)
- Answer
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1. \(a_{9}=2,304 .\) The general term is \(a_{n}=9(2)^{n-1}\).
3. \(a_{15}=-\frac{2}{19,683} .\) The general term is \(a_{n}=-486\left(-\frac{1}{3}\right)^{n-1}\).
5. \(a_{10}=0.000000001 .\) The general term is \(a_{n}=(0.1)^{n-1}\).
Exercise \(\PageIndex{26}\) Find the Sum of the First \(n\) terms of a Geometric Sequence
In the following exercises, find the sum of the first fifteen terms of each geometric sequence.
- \(8,24,72,216,648,1944, \dots\)
- \(7,14,28,56,112,224, \dots\)
- \(-6,12,-24,48,-96,192, \dots\)
- \(-4,12,-36,108,-324,972, \ldots\)
- \(81,27,9,3,1, \frac{1}{3}, \ldots\)
- \(256,64,16,4,1, \frac{1}{4}, \frac{1}{16}, \dots\)
- Answer
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1. \(57,395,624\)
3. \(-65,538\)
5. \(\frac{7,174,453}{59,049} \approx 121.5\)
Exercise \(\PageIndex{27}\) Find the Sum of the First \(n\) terms of a Geometric Sequence
In the following exercises, find the sum of the geometric sequence.
- \(\sum_{i=1}^{15}(2)^{i}\)
- \(\sum_{i=1}^{10}(3)^{i}\)
- \(\sum_{i=1}^{9} 4(2)^{i}\)
- \(\sum_{i=1}^{8} 5(3)^{i}\)
- \(\sum_{i=1}^{10} 9\left(\frac{1}{3}\right)^{i}\)
- \(\sum_{i=1}^{15} 4\left(\frac{1}{2}\right)^{i}\)
- Answer
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1. \(65,534\)
3. \(4088\)
5. \(\frac{29,524}{6561} \approx 4.5\)
Exercise \(\PageIndex{28}\) Find the Sum of an Infinite Geometric Series
In the following exercises, find the sum of each infinite geometric series.
- \(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}+\frac{1}{729}+\ldots\)
- \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\ldots\)
- \(6-2+\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+\ldots\)
- \(-4+2-1+\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\dots\)
- \(6+12+24+48+96+192+\dots\)
- \(5+15+45+135+405+1215+\ldots\)
- \(1,024+512+256+128+64+32+\ldots\)
- \(6,561+2187+729+243+81+27+\dots\)
- Answer
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1. \(\frac{3}{2}\)
3. \(\frac{9}{2}\)
5. no sum as \(r≥1\)
7. \(2,048\(
Exercise \(\PageIndex{29}\) Find the Sum of an Infinite Geometric Series
In the following exercises, write each repeating decimal as a fraction.
- \(0 . \overline{3}\)
- \(0 . \overline{6}\)
- \(0 . \overline{7}\)
- \(0 . \overline{2}\)
- \(0 . \overline{45}\)
- \(0 . \overline{27}\)
- Answer
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1. \(\frac{1}{3}\)
3. \(\frac{7}{9}\)
5. \(\frac{5}{11}\)
Exercise \(\PageIndex{30}\) Apply Geometric Sequences and Series in the Real World
In the following exercises, solve the problem.
- Find the total effect on the economy of each government tax rebate to each household in order to stimulate the economy if each household will spend the indicated percent of the rebate in goods and services.
| Tax rebate to each household | Percent spent on goods and services | Total Effect on the economy |
|---|---|---|
| a. $\(1,000\) | \(85\)% | |
| b. $\(1,000\) | \(75\)% | |
| c. $\(1,500\) | \(90\)% | |
| d. $\(1,500\) | \(80\)% |
2. New grandparents decide to invest $\(100\) per month in an annuity for their grandchild. The account will pay \(6\)% interest per year which is compounded monthly (\(12\) times a year). How much will be in the child’s account at their twenty-first birthday?
3. Berenice just got her first full-time job after graduating from college at age \(30\). She decided to invest $\(500\) per quarter in an IRA (an annuity). The interest on the annuity is \(7\)% which is compounded quarterly (\(4\) times a year). How much will be in the Berenice’s account when she retires at age \(65\)?
4. Alice wants to purchase a home in about five years. She is depositing $\(500\) a month into an annuity that earns \(5\)% per year that is compounded monthly (\(12\) times a year). How much will Alice have for her down payment in five years?
5. Myra just got her first full-time job after graduating from college. She plans to get a master’s degree, and so is depositing $\(2,500\) a year from her year-end bonus into an annuity. The annuity pays \(6.5\)% per year and is compounded yearly. How much will she have saved in five years to pursue her master’s degree?
- Answer
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1. a. $\(6666.67\) b. $\(4000\) c. $\(15,000\) d. $\(7500\)
3. $\(295,581.88\)
5. $\(14,234.10\)
Exercise \(\PageIndex{31}\) Writing Exercises
- In your own words, explain how to determine whether a sequence is geometric.
- In your own words, explain how to find the general term of a geometric sequence.
- In your own words, explain the difference between a geometric sequence and a geometric series.
- In your own words, explain how to determine if an infinite geometric series has a sum and how to find it.
- Answer
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2. Answers will vary.
4. Answers will vary
Self Check
- After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
- What does this checklist tell you about your mastery of this section? What steps will you take to improve?


5.2: Complex Solution
The variable \(x(t)\) is the displacement of the particle, so it ought to be real. However, a good way to solve the damped harmonic oscillator equation is to generalize \(x(t)\) to complex values. In other words, we convert the harmonic oscillator equation into a complex ODE: \[\frac{d^2 z}{dt^2} + 2\gamma \frac{dz}{dt} + \omega_0^2 z(t) = 0, \quad z(t) \in \mathbb{C}.\] The parameter-counting rule for real ODEs (see Chapter 2) generalizes to complex ODEs, but the parameters are now complex numbers. As the complex damped harmonic oscillator equation is a second-order ODE, its general solution must have two complex free parameters.
Let us now figure out how to obtain the general solution to the complex damped harmonic oscillator equation. Then we will see how to use it to solve the real problem.
Complex ansatz
To derive the general solution, first note that the damped harmonic oscillator equation is linear. If we have two solutions \(z_1(t)\) and \(z_2(t)\) , then any superposition \[\psi_1 \, z_1(t) + \psi_2 \,z_2(t),\quad \mathrm{where}\;\, \psi_1, \psi_2 \in \mathbb{C}\] is also a solution. This can be verified by direct substitution into the ODE.
Therefore, a good strategy for obtaining the general solution is to find two specific solutions and superpose them. The two coefficients, \(\psi_1\) and \(\psi_2\) , would then serve as the general solution’s two free parameters.
We can make a guess (or an ansatz ) for a specific solution: \[z(t) = e^{-i\omega t}.\] Here, \(\omega\) is a constant to be determined (which could be complex). The first and second derivatives are: \[\begin{align} \frac{dz}{dt} &= -i\omega\, e^{-i\omega t} \\ \frac{d^2z}{dt^2} &= -\omega^2\, e^{-i\omega t}\end{align}\] Substituting these into the differential equation gives: \[\left[-\omega^2 - 2i\gamma \omega + \omega_0^2 \right] e^{-i\omega t} = 0.\] This equation holds for all \(t\) if and only if the complex second-order polynomial on the left-hand side is zero: \[-\omega^2 - 2i\gamma \omega + \omega_0^2 = 0.\] The solutions for \(\omega\) can be obtained from the quadratic formula: \[\omega = -i\gamma \pm \sqrt{\omega_0^2 - \gamma^2}.\] Hence, we have found specific solutions that involve complex frequencies: \[z(t) = \exp\big(-i\omega_\pm t\big), \;\;\mathrm{where}\;\; \omega_\pm = -i\gamma \pm \sqrt{\omega_0^2 - \gamma^2}.\] For each value of \(\gamma\) and \(\omega_0\) , both \(\omega_+\) and \(\omega_-\) yield valid specific solutions.
Complex frequencies
What does it mean to have an oscillation with a complex frequency? If we write the real and imaginary parts of the frequency as \(\omega = \omega_R + i \omega_I\) , then \[z(t) = e^{-i\omega t} = e^{\omega_I t} \; e^{-i\omega_R t}.\]
If both \(\omega_R\) and \(\omega_I\) are non-zero, this describes a spiral trajectory in the complex plane whose magnitude either increases or decreases with time, depending on the sign of \(\omega_I\) . To see this explicitly, we can write \[z(t) = e^{\omega_I t} \; e^{-i\omega_R t} = R(t)\, e^{i\theta(t)}, \;\;\mathrm{where}\;\,\begin{cases}\displaystyle R(t) &= e^{\omega_I t}, \\ \displaystyle \theta(t) &= -\omega_R t.\end{cases}\] The real part of \(\omega\) determines the oscillation frequency, and the imaginary part determines whether the amplitude grows with time (amplification) or shrinks with time (damping). A positive imaginary part implies amplification, and a negative imaginary part implies damping; zero imaginary part (i.e., a real frequency) implies constant-amplitude oscillation.
Now let’s look at the complex frequencies appearing in the specific solutions to the damped harmonic oscillator: \[\omega_\pm = -i\gamma \pm \sqrt{\omega_0^2 - \gamma^2}.\] In the plot below, you can see how the position of \(\omega_\pm\) in the complex plane depends on the values of \(\gamma\) and \(\omega_0\) :
In particular, note the following features:
- For \(\gamma = 0\) (zero damping), the two frequencies are both real, and take the values \(\pm \omega_0\) . This corresponds to simple harmonic oscillation at the oscillator’s natural frequency.
- If we increase \(\gamma\) from zero with \(\omega_0\) fixed, both \(\omega_+\) and \(\omega_-\) move downwards in the complex plane, along a circular arc. Since the imaginary part of the frequencies are negative, the particle undergoes damped oscillation. This is called under-damped motion .
- At \(\gamma = \omega_0\) , the frequencies meet along the imaginary axis. This is the case of critical damping , which we will discuss in Section 5.3.
- For \(\gamma > \omega_0\) , the two frequencies move apart along the imaginary axis. Purely imaginary frequencies correspond to a trajectory that decays without oscillating. This is called over-damped motion , which we will discuss in Section 5.3.


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Micronutrients are essential for adequate neurotransmission.
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Diet modifications can support brain health in individuals of all ages.
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Metabolic syndrome can lead to cognitive decline.
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Diet modifications can slow the decline of some neurodegenerative conditions, including Parkinson’s disease, Alzheimer’s disease, and multiple sclerosis.
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Long-term effective diet modifications cannot be made in a single health care encounter.